I know this is a basic question, but given a morphism $f:M \rightarrow N$, where $M$ and $N$ are say, monoids. For any $ \{a,b\} \in M$, we have $f(ab)=f(a)f(b)$ from the definition of a morphism.
Now, let $\{c,d\} \in N$, and $f^{-1}$ as the inverse of $f$, does the statement below follow in general ?
$$ f^{-1}(cd)=f^{-1}(c)f^{-1}(d) $$